The Formula for a Perfect Square Trinomial

Oct 13, 2025 by miodeya.com 38 views

Introduction

In algebra, a perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. It has a specific form, which is obtained by squaring a binomial. In this lesson, we will learn how to determine if a given trinomial is a perfect square trinomial or not. We will use the trinomial $x^2 + 4x - 7$ as an example to illustrate the process.

What is a Perfect Square Trinomial?

A perfect square trinomial is a quadratic expression that can be written in the form $(x + a)^2$ or $(x - a)^2$ , where $a$ is a constant. When we expand the square of a binomial, we get a trinomial of the form $x^2 + 2ax + a^2$ . This is the form of a perfect square trinomial.

The Formula for a Perfect Square Trinomial

The formula for a perfect square trinomial is:

$(x + a)^2 = x^2 + 2ax + a^2$

$(x - a)^2 = x^2 - 2ax + a^2$

Determining if a Trinomial is a Perfect Square Trinomial

To determine if a trinomial is a perfect square trinomial, we need to check if it can be written in the form $(x + a)^2$ or $(x - a)^2$ . We can do this by comparing the given trinomial with the formula for a perfect square trinomial.

Example: Is the Trinomial $x^2 + 4x - 7$ a Perfect Square Trinomial?

Let's use the trinomial $x^2 + 4x - 7$ as an example. We need to determine if it is a perfect square trinomial or not.

Step 1: Compare the Trinomial with the Formula

We compare the given trinomial $x^2 + 4x - 7$ with the formula for a perfect square trinomial:

$(x + a)^2 = x^2 + 2ax + a^2$

We can see that the given trinomial has a term $4x$ , which is twice the value of $a$ . This suggests that the trinomial may be a perfect square trinomial.

Step 2: Check if the Trinomial Can be Written in the Form $(x + a)^2$

We need to check if the trinomial $x^2 + 4x - 7$ can be written in the form $(x + a)^2$ . We can do this by trying to find a value of $a$ that makes the trinomial equal to $(x + a)^2$ .

Step 3: Expand the Square of a Binomial

We expand the square of a binomial $(x + a)^2$ to get:

$(x + a)^2 = x^2 + 2ax + a^2$

We can see that the trinomial $x^2 + 4x - 7$ has a term $-7$ , which is not equal to $a^2$ . This suggests that the trinomial is not a perfect square trinomial.

Step 4: Check if the Trinomial Can be Written in the Form $(x - a)^2$

We need to check if the trinomial $x^2 + 4x - 7$ can be written in the form $(x - a)^2$ . We can do this by trying to find a value of $a$ that makes the trinomial equal to $(x - a)^2$ .

Step 5: Expand the Square of a Binomial

We expand the square of a binomial $(x - a)^2$ to get:

$(x - a)^2 = x^2 - 2ax + a^2$

We can see that the trinomial $x^2 + 4x - 7$ has a term $4x$ , which is twice the value of $a$ . This suggests that the trinomial may be a perfect square trinomial.

Step 6: Check if the Trinomial is a Perfect Square Trinomial

We need to check if the trinomial $x^2 + 4x - 7$ is a perfect square trinomial. We can do this by checking if the trinomial can be written in the form $(x - a)^2$ .

Step 7: Find the Value of $a$

We need to find the value of $a$ that makes the trinomial $x^2 + 4x - 7$ equal to $(x - a)^2$ . We can do this by equating the two expressions and solving for $a$ .

Step 8: Solve for $a$

We equate the two expressions:

$x^2 + 4x - 7 = (x - a)^2$

We expand the square of the binomial:

$x^2 + 4x - 7 = x^2 - 2ax + a^2$

We equate the coefficients of the terms:

$4x = -2ax$

We solve for $a$ :

$a = -2$

Step 9: Check if the Trinomial is a Perfect Square Trinomial

We need to check if the trinomial $x^2 + 4x - 7$ is a perfect square trinomial. We can do this by checking if the trinomial can be written in the form $(x - a)^2$ .

Step 10: Write the Trinomial in the Form $(x - a)^2$

We write the trinomial $x^2 + 4x - 7$ in the form $(x - a)^2$ :

$x^2 + 4x - 7 = (x - 2)^2$

We can see that the trinomial $x^2 + 4x - 7$ is indeed a perfect square trinomial.

Conclusion

In this lesson, we learned how to determine if a given trinomial is a perfect square trinomial or not. We used the trinomial $x^2 + 4x - 7$ as an example to illustrate the process. We compared the trinomial with the formula for a perfect square trinomial, expanded the square of a binomial, and solved for the value of $a$ . We found that the trinomial $x^2 + 4x - 7$ is indeed a perfect square trinomial.

Practice Problems

Is the trinomial $x^2 - 6x + 8$ a perfect square trinomial?
Is the trinomial $x^2 + 2x - 3$ a perfect square trinomial?
Is the trinomial $x^2 - 4x - 5$ a perfect square trinomial?

Answer Key

No
No
No
Frequently Asked Questions (FAQs) about Perfect Square Trinomials ====================================================================

Q: What is a perfect square trinomial?

A: A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. It has a specific form, which is obtained by squaring a binomial.

Q: How do I determine if a trinomial is a perfect square trinomial?

A: To determine if a trinomial is a perfect square trinomial, you need to check if it can be written in the form $(x + a)^2$ or $(x - a)^2$ . You can do this by comparing the given trinomial with the formula for a perfect square trinomial.

Q: What is the formula for a perfect square trinomial?

A: The formula for a perfect square trinomial is:

$(x + a)^2 = x^2 + 2ax + a^2$

$(x - a)^2 = x^2 - 2ax + a^2$

Q: How do I expand the square of a binomial?

A: To expand the square of a binomial, you need to multiply the binomial by itself. For example, to expand $(x + a)^2$ , you would multiply $(x + a)$ by $(x + a)$ .

Q: What is the value of $a$ in a perfect square trinomial?

A: The value of $a$ in a perfect square trinomial is the value that makes the trinomial equal to $(x + a)^2$ or $(x - a)^2$ .

Q: How do I find the value of $a$ in a perfect square trinomial?

A: To find the value of $a$ in a perfect square trinomial, you need to equate the given trinomial with the formula for a perfect square trinomial and solve for $a$ .

Q: What are some examples of perfect square trinomials?

A: Some examples of perfect square trinomials include:

$(x + 2)^2 = x^2 + 4x + 4$
$(x - 3)^2 = x^2 - 6x + 9$
$(x + 1)^2 = x^2 + 2x + 1$

Q: How do I write a trinomial in the form $(x + a)^2$ or $(x - a)^2$ ?

A: To write a trinomial in the form $(x + a)^2$ or $(x - a)^2$ , you need to find the value of $a$ that makes the trinomial equal to $(x + a)^2$ or $(x - a)^2$ .

Q: What are some common mistakes to avoid when working with perfect square trinomials?

A: Some common mistakes to avoid when working with perfect square trinomials include:

Not checking if the trinomial can be written in the form $(x + a)^2$ or $(x - a)^2$
Not expanding the square of a binomial correctly
Not solving for the value of $a$ correctly

Q: How do I practice working with perfect square trinomials?

A: To practice working with perfect square trinomials, you can try the following:

Work through examples of perfect square trinomials
Practice expanding the square of a binomial
Practice solving for the value of $a$ in a perfect square trinomial
Try to write a trinomial in the form $(x + a)^2$ or $(x - a)^2$

Conclusion

In this article, we have answered some frequently asked questions about perfect square trinomials. We have covered topics such as the definition of a perfect square trinomial, how to determine if a trinomial is a perfect square trinomial, and how to expand the square of a binomial. We have also provided some examples of perfect square trinomials and some common mistakes to avoid when working with them.